I'm having trouble understanding a small detail from a paper I'm reading about Roth's theorem. Below is the context.
Let $A \subset \mathbb{Z}_{N}$ be such that $\left|\sum_{x \in A}{\exp\left(\frac{2\pi i rx}{N}\right)} \right| \geq 3 \gamma N$ for some $r \neq 0$, and $\gamma$ is such that $1/\gamma$ is a positive integer $n$. Assume that $N$ is a large prime number (in particular so that $r$ and $N$ are coprime).
Now, divide the unit circle in the complex in $n$ equal parts and let $P_{k} = \left\{x \in \mathbb{Z}_{N}: \exp\left(\frac{2\pi i rx}{N}\right) \in I_{k}\right\}$, where $r \neq 0$, and $k$ is a fixed number $k \in \left\{1,\ldots,n\right\}$. Why does it follow that $P_{k}$ is an arithmetic progression (with difference $r^{-1}$, the inverse of $r$ in $\mathbb{Z}_{N}$)?
Thanks!